Minimize trace of $A$ given that $A-N$ is positive semi-definite.

Question

I'm currently working on a problem in harmonic analysis and came across the following linear algebra problem:

Suppose $A$ and $N$ are two positive semi-definite operators on $\mathbb C^4$ such that $\text{trace}(N)=1$, I want to minimize $\text{trace(A)}$ subject to the constraint that $A-N$ is positive semi-definite. Is there a general method to trace optimization for operators on $\mathbb C^n$?

Answer 1

Since $A-N$ is psd, $\text{trace}(A-N)\geq 0$. Thus $\text{trace}(A)\geq \text{trace}(N)=1$. Equality holds if and only if $A=N$, since any non-zero psd matrix has strictly positive trace. (Thanks to user1551 for pointing out the last sentence.)